SPCAvRP {SPCAvRP}

Computes the leading eigenvector using the SPCAvRP algorithm

Description

Computes l-sparse leading eigenvector of the sample covariance matrix, using A x B random axis-aligned projections of dimension d. For the multiple component estimation use SPCAvRP_subspace or SPCAvRP_deflation.

Usage

SPCAvRP(data, cov = FALSE, l, d = 20, A = 600, B = 200, 
center_data = TRUE, parallel = FALSE, 
cluster_type = "PSOCK", cores = 1, machine_names = NULL)

Arguments

data

Either the data matrix (p x n) or the sample covariance matrix (p x p).

cov

TRUE if data is given as a sample covariance matrix.

l

Desired sparsity level in the final estimator (see Details).

d

The dimension of the random projections (see Details).

A

Number of projections over which to aggregate (see Details).

B

Number of projections in a group from which to select (see Details).

center_data

TRUE if the data matrix should be centered (see Details).

parallel

TRUE if the selection step should be computed in parallel by uses package "parallel".

cluster_type

If parallel == TRUE, this can be "PSOCK" or "FORK" (cf. package "parallel").

cores

If parallel == TRUE and cluster_type == "FORK", number of cores to use.

machine_names

If parallel == TRUE, the names of the computers on the network.

Details

This function implements the SPCAvRP algorithm for the principal component estimation (Algorithm 1 in the reference given below).

If the true sparsity level k is known, use l = k and d = k.

If the true sparsity level k is unknown, l can take an array of different values and then the estimators of the corresponding sparsity levels are computed. The final choice of l can then be done by the user via inspecting the explained variance computed in the output value or via inspecting the output importance_scores. The default choice for d is 20, but we suggest choosing d equal to or slightly larger than l.

It is desirable to choose A (and B = ceiling(A/3)) as big as possible subject to the computational budget. In general, we suggest using A = 300 and B = 100 when the dimension of data is a few hundreds, while A = 600 and B = 200 when the dimension is on order of 1000.

If center_data == TRUE and data is given as a data matrix, the first step is to center it by executing scale(data, center_data, FALSE), which subtracts the column means of data from their corresponding columns.

If parallel == TRUE, the parallelised SPCAvRP algorithm is used. We recommend to use this option if p, A and B are very large.

Value

Returns a list of three elements:

vector

A matrix of dimension p x length(l) with columns as the estimated eigenvectors of sparsity level l.

value

An array with length(l) eigenvalues corresponding to the estimated eigenvectors returned in vector.

importance_scores

An array of length p with importance scores for each variable 1 to p.

Author(s)

Milana Gataric, Tengyao Wang and Richard J. Samworth

References

Milana Gataric, Tengyao Wang and Richard J. Samworth (2018) Sparse principal component analysis via random projections https://arxiv.org/abs/1712.05630

Examples

p <- 100  # data dimension
k <- 10   # true sparsity level
n <- 1000 # number of observations
v1 <- c(rep(1/sqrt(k), k), rep(0,p-k)) # true principal component
Sigma <- 2*tcrossprod(v1) + diag(p)    # population covariance
mu <- rep(0, p)                        # population mean
loss = function(u,v){ 
  # the loss function
  sqrt(abs(1-sum(v*u)^2))
}
set.seed(1)
X <- mvrnorm(n, mu, Sigma) # data matrix

spcavrp <- SPCAvRP(data = X, cov = FALSE, l = k, d = k, A = 200, B = 70)
spcavrp.loss <- loss(v1,spcavrp$vector)
print(paste0("estimation loss when l=d=k=10, A=200, B=70: ", spcavrp.loss))

##choosing sparsity level l if k unknown:
#spcavrp.choosel <- SPCAvRP(data = X, cov = FALSE, l = c(1:30), d = 15, A = 200, B = 70)
#plot(1:p,spcavrp.choosel$importance_scores,xlab='variable',ylab='w',
#     main='choosing l when k unknown: \n importance scores w')
#plot(1:30,spcavrp.choosel$value,xlab='l',ylab='Var_l',
#     main='choosing l when k unknown: \n explained variance Var_l')